CSIS 3510 Computer Organization and ArchitectureDe’Morgan’s TheoremUsing De’Morgan’s TheoremExtracting a function from a truth table and convertingTwo-Bit DecoderTwo-Bit Decoder DiagramAlgebraic Reduction of Boolean Functions7-Segment Display Decoder7-Segment Display Truth TableExtracting and Minimizing Segment FunctionsSlide 11Slide 127-Segment Display Decoder DiagramConclusionCSIS 3510 Computer Organization and Architecture•Topics covered in this lecture:–Review of De’Morgan’s Theorem–Using De’Morgan’s Theorem–Building a 2-bit decoder–Algebraic Reduction of Boolean Expressions–7-segment decoderDe’Morgan’s Theorem•On of the most useful principles in boolean algebra is De’Morgan’s Theorem, which allows one to switch between ANDs and NORs and ORs and NANDs.•Remember, we want to design circuits using AND and OR, but then implement them using NAND and NOR (AND transistor bleed-through problem, and manufacturing layering minimization)•NOT terms or Inverted terms are represented with a line over the terms•AB = A + B •A + B = AB•We demonstrated the validity of DeMorgan’s by Perfect Induction, using a truth table.Using De’Morgan’s TheoremTo convert A+B into a form that can be implemented using a NAND gate follow these steps:•1. Double Complement the term A+B = A+B•2. Use DeMorgan’s to distribute one of the complementsA+B = A BThe equation is now a NAND of the complemented inputs.To convert AB into a form that can be implemented using a NOR gate follow these steps:•1. Double Complement the term AB = AB•2. Use DeMorgan’s to distribute one of the complementsAB = A + BThe equation is now a NOR of the complemented inputs.Extracting a function from a truth table and convertingA B Output0 0 00 1 11 0 0 Out = A B1 1 0 DoubleC A BDeM A + BSimplify A + BTwo-Bit DecoderA B F0F1F2F30 0 1 0 0 00 1 0 1 0 01 0 0 0 1 01 1 0 0 0 1F0 = A B = A B = A+B = A+BF1 = A B = A B = A+B = A+BF2 = A B = A B = A+B = A+BF3 = A B = A B = A+BTwo-Bit Decoder DiagramA A B BF0F1F2F3Algebraic Reduction of Boolean Functions•Algebraic reduction is used to minimize a function extracted from a truth table or other source.•Principle is to look for terms where a single variable is present in both complemented and positive form. Those variables can be deleted from those terms.•Note: terms can be used over in multiple minimizations.•Examples•A B + A B = B, A B + A B = A•A B C + A B C = A B•A B C + A B C + A B C + A B C A B A B A7-Segment Display Decoder•7 Segment Display:•Displaying 0, 1, 2, 301234567-Segment Display Truth Table(Displaying values from 0-3 only)A B F0 F1 F2 F3 F4 F5 F6 0 0 1 1 1 1 1 1 00 1 0 1 1 0 0 0 01 0 1 1 0 1 1 0 11 1 1 1 1 1 0 0 1Extracting and Minimizing Segment FunctionsF0 =F1 =F2 =F3 =F4 =F5 =F6 =Extracting and Minimizing Segment FunctionsF0 =F1 =F2 =F3 =F4 =F5 =F6 =Extracting and Minimizing Segment FunctionsF0 =F1 =F2 =F3 =F4 =F5 =F6 =7-Segment Display Decoder DiagramConclusion•Topics covered in this lecture:–Review of De’Morgan’s Theorem–Using De’Morgan’s Theorem–Building a 2-bit decoder–Algebraic Reduction–7-segment